You can try to define the determinant of an $n \times n$ matrix with entries in a bipermutative (or symmetric bimonoidal) category $R$ by an analogue of the usual signed sum of $n$-fold products. However, it will usually not be a monoidal functor, and the inclusion $BGL_1(R) \to K(R)$ does generally not admit a retraction, which you might expect to get from a determinant. There is a counterexample in

- Christian Ausoni, Bjørn Ian Dundas, John Rognes, _[Divisibility of the Dirac magnetic monopole as a two-vector bundle over the three-sphere](https://www.math.uni-bielefeld.de/documenta/vol-13/22.html)_, Doc. Math. **13** (2008), 795–801.

and a discussion of what more might be needed (to circumvent this obstruction) in

- Thomas Kragh, _[Orientations on 2-vector bundles and determinant gerbes](https://doi.org/10.7146/math.scand.a-15482)_, Math. Scand. **113** (2013), no. 1, 63–82.